The present invention relates generally to chromatographic techniques such as high speed liquid chromatography and gas chromatography, and more particularly to methods and apparatuses for analyzing a multichannel chromatogram which is detected by a diode array detector or the like.
A large number of approaches have been proposed for separating or resolving overlapping peaks on a multichannel chromatogram. The multichannel chromatogram refers to a type of a chromatogram having three-dimensional information composed of an absorbance component, a time component, and a wavelength component. Representatives of such approaches include factor analysis and principal component analysis which are techniques of multivariate analysis (see, for example, U.S. Pat. No. 4,807,148 issued to J. K. Strasters et al.; H. R. Keller et at, Journal of Liquid Chromatography, No. 12, pp. 3-22 (1988); J. Craig Hamilton et al., Chemometrics and Intelligent Laboratory Systems, No. 12, pp. 209-224 (1992); Journal of Chemometrics, No. 4, pp. 1-13 (1990), and so on).
A non-linear least-squares analysis has also been proposed as another approach for analyzing a multichannel chromatogram (see, for example, Itoh et al, Abstracts of Seventh Liquid Chromatography Conference, pp. 5 (1986); Itoh et al, Abstracts of 22th Applied Spectrometry Tokyo Conference, pp. 141 (1987); and Itoh et at, Abstracts of 19th HPLC Research Conversation, pp. 30 (1988)). This approach has been developed from a method for fitting overlapping peaks on a multichannel chromatogram obtained from GS/MS using the Gaussian (normal distribution function), which was proposed by F. J. Knorr, by substituting EMG (Exponentially Modified Gaussian) as a more realistic model function for the former Gaussian. This approach will be explained here in detail since it will facilitate the understanding of later descriptions.
Multichannel chromatogram data gathered from a diode array detector is given by the following matrix Dij (1):
D=Rxc2x7Axc2x7S=Rxc2x7Xxe2x80x83xe2x80x83(1)
where i represents a time index, j a wavelength index, k a component index, Dij the absorbance, Rik a normalized retention waveform (chromatogram magnitude), Akk a quantitative factor relative to the concentration of a k-component, Skj a normalized spectral intensity, and Xkj a quantitative spectral intensity multiplied with the factor Akk relative to the concentration of a k-component. It should be noted that the wavelength index j is used as a n index of the m/z value when a mass spectrometer is used.
Here, a trial matrix Rxe2x80x2ik is introduced in place of the chromatogram intensity matrix Rik. From the equation (1), a trial spectral intensity matrix Xxe2x80x2jk can be computed from the trial matrix Rxe2x80x2kl and the matrix Dij containing the measured data. Consequently, a trial data matrix Dxe2x80x2ij can be obtained as expressed by the following equation (2):
Dxe2x80x2=Rxe2x80x2xc2x7(Rxe2x80x2Txc2x7Rxe2x80x2)xe2x88x921xc2x7Rxe2x80x2xc2x7TDxe2x80x83xe2x80x83(2)
where T in Rxe2x80x2T means a transpose matrix, and xe2x88x921 in (Rxe2x80x2Txc2x7Rxe2x80x2)xe2x88x921 means that the associated matrix is an inverse matrix.
The least-squares method determines parameters so as to minimize Dij-Dxe2x80x2ij and provides the best-fit Rxe2x80x2ik for each component. In this way, the respective component matrices Rxe2x80x2ik and Xxe2x80x2ik can be separated or resolved from the matrix Dij including data on overlapping peaks.
This approach utilizes EMG which can also represent a asymmetric tailing peak as the Rxe2x80x2ik matrix (the following equation (3)).
Rxe2x80x2ik=1/(xcfx84k"sgr"k(2xcfx80)xc2xd)∫ot1 exp[xe2x88x92(tixe2x88x92tRKxe2x88x92txe2x80x2)2]/2"sgr"k2xe2x88x92txe2x80x2/xcfx84k)]dtxe2x80x2xe2x80x83xe2x80x83(3)
where tRK is retention time for a k-component, "sgr"K standard deviation for the k-component, and xcfx84K the time constant for the k-component.
JP-A-60-2447 describes a chromatographic quantitative analyzing method and apparatus which detect temporal changes in absorbance at multiple wavelengths as three-dimensional information to perform quantitative analysis. The chromatographic quantitative analyzing method and apparatus employ functions f1(xcex) and f2(xcex) which represent two previously measured two-dimensional standard spectra. Then, an equation is obtained for a two-dimensional synthesized component function fs(xcex) of a measured sample having overlapping peaks using the above-mentioned functions f1(xcex) and f2(xcex). The overlapping peaks are resolved by the obtained equation.
Other approaches utilizing deconvolution have also been proposed as a method for analyzing an ordinary chromatogram, i.e., a chromatogram having two-dimensional information (composed of an absorbance component and a time component). See, for example, U.S. Pat. No. 4,941,101 issued to Paul Benjamine Crilly; Paul Benjamine Crilly, IEEE Transaction on Instrumentation and Measurement, No. 40, pp. 558-562 (1991); and Journal of Chemometrics, No. 5, pp. 85-95 (1991).
Here, the convolution is defined. Original data is detected by a detector, and dispersed by an inherent device function h(t) (dispersion function(instrument function)) which represents the detection characteristics of the detector. The deconvolution is determined to be the processing for removing a dispersion portion of the data by the device function h(t) from the dispersion data. An equation defining the deconvolution is given in the following equation (4):
D(t)=∫xe2x88x92∞∞h(txe2x80x2)d(txe2x88x92txe2x80x2)dtxe2x80x2=h(t)*d(t)xe2x80x83xe2x80x83(4)
where D(t) is a detected waveform, d(t) an original waveform, and h(t) a dispersion function.
For the convolution applied to analysis on a chromatogram having two-dimensional information, several approaches have been proposed for promoting the convergence. Specifically, these approaches have been proposed principally relying on iteration methods, and include the Gaussian elimination which performs an inverse matrix operation, as well as Jacobi""s method, Gauss-Seidel""s method, Fourier Transform method, Van Cittert""s method, Constrained Iterative method, Jansson""s method, Gold""s ratio method, and so on. For details of these methods, see xe2x80x9cWaveform Data Processing for Scientific Measurementsxe2x80x9d, edited by Shigeo Minami, published by CQ Editorial, pp. 122-139 (1986); and P. A. Jansson, xe2x80x9cDeconvolution with Applications in Spectroscopy, New York, Academic (1984).
Also, a method based on factor analysis for separating or resolving overlapping peaks on a multichannel chromatogram is introduced in detail, for example, by Edmund R. Malinowski, xe2x80x9cFactor Analysis in Chemistryxe2x80x9d, John Wiley and Sons, Inc. (1991).
This factor analysis based method is multivariate analysis, and its basic thinking is that a data matrix D is modeled as a product of a spectral matrix X and an elution profile matrix Y, as represented by the following equations (5) and (6). It should be noted however that the equation (6) defines that each component k is normalized to a peak area of one. However, since the data matrix D cannot be uniformly resolved from a mathematical point of view, several rational constraints are provided to solve the problem.
D=XYxe2x80x83xe2x80x83(5)
                                          ∑            j                    ⁢                      Y            kj                          =                              Y                          k              -                                =          1                                    (        6        )            
where
Dij: Signal Magnitude;
Xik: Spectral Intensity;
Ykj: Elution Profile;
i: Channel Index;
k: Component Index;
j; Time Index.
Generally, an elgenvalue problem is solved for the data matrix D, the number n of components is determined by principal component analysis, an abstract elution profile matrix V having a characteristic vector as its element is transformed by a matrix T, and thus a physically meaningful elution profile matrix Y is obtained. As the matrix Y is determined, the spectral matrix X can be computed from the data matrix D by the following equations (7)-(15).
More specifically, solving the eigenvalue problem, the data matrix D is represented by a product of an abstract spectral matrix U and the abstract elution profile matrix V as given by the following equation (7):
D=UVxe2x80x83xe2x80x83(7)
Here, a characteristic vector of a product Z=Dxcfx84D, i.e., a product of a transposed matrix Dxcfx84 of the matrix D with the matrix D is {right arrow over (Vk)}. This vector {right arrow over (Vk)} is a kxe2x80x2th row vector of the matrix V given by the following equation (8):                     V        =                  (                                                                      V1                  →                                                                                    ⋮                                                                                      Vk                  →                                                                                    ⋮                                                                                      Vn                  →                                                              )                                    (        8        )            
Also, the relationship between a characteristic value xcex6k, the characteristic vector {right arrow over (Vk)}, and the matrix Z is given by the following equation (9):
Z{right arrow over (Vk)}=xcex6k{right arrow over (Vk)}xe2x80x83xe2x80x83(9)
Further, the vector {right arrow over (Vk)} is such that its sum of squares is normalized to one as given by the following equation (10):                                           ∑            j                    ⁢                                    (                              V                kj                            )                        2                          =                                            "LeftBracketingBar"                                                V                                      k                    -                                                  →                            "RightBracketingBar"                        2                    =          1                                    (        10        )            
The matrix V can be transformed to the matrix Y by a transformation matrix T as shown in the following equation (11);
Y=TVxe2x80x83xe2x80x83(11)
where the matrix T is an nxn matrix.
The matrix T serves to perform an oblique rotation and a transformation for transforming the area of each raw vector in the matrix Y to one.
The matrix Y is represented by the following equation (12), where any component {right arrow over (Yk)} of the matrix Y satisfies the following equation (13):                     Y        =                  (                                                                      Y1                  →                                                                                    ⋮                                                                                      Yk                  →                                                                                    ⋮                                                                                      Yn                  →                                                              )                                    (        12        )                                                                    ∑              j                        n                    ⁢                      Y            kj                          =                              Y                          k              -                                =          1                                    (        13        )            
Also, the matrix X is obtained from the matrices D and Y as shown in the following equation (14):
X=DYxcfx84(YYxcfx84)xe2x88x921xe2x80x83xe2x80x83(14)
The matrices X, Y, U, and V have the relationship represented by the following equation (15):
D=XY=(UTxe2x88x921)(TV)xe2x80x83xe2x80x83(15)
The transformation matrix T, however, cannot be easily determined. Therefore, approaches as follows have been proposed for determining the transformation matrix T.
1. Methods with Known Spectrum:
1-1. TTFA (Target Transformation Factor Analysis): A method for determining a transformation matrix T such that a known spectral waveform is obtained from an abstract spectral matrix.
1-2. RAFA (Rank Annihilation Factor Analysis): A method for obtaining a known data matrix from standard forms of respective components and subtracting the components one by one from the obtained data matrix.
1-3. GRATA (Generalized Rank Annihilation Factor Analysis): While RAFA (1-2) requires a data matrix having columns with one component, this approach enables curve resolution to be performed using a data matrix obtained from a standard mixed sample.
2. Modeling Methods with Unknown Spectrum:
2-1. Gaussian Non-linear Least-Squares Method: A method for performing a modeling on assumption that an elution profile is Gaussian, and the model is fit by the non-linear least-squares method.
2-2. Non-linear Least-Squares Method Using the Aforementioned EMG (Exponentially Modified Gaussian): A method identical to the above method 2-1 except for employing EMG which can represent asymmetric peaks in place of Gaussian.
3. Self Modeling Method with Unknown Spectrum:
3-1. ITTFA (Iterative Target Transformation Factor Analysis): A method which initially introduces a test vector having a pulsatile elution profile and gradually adjusts it to approach to a true elution profile.
3-2. EFA (Evolving Factor Analysis): A method which plots changes in the characteristic value along the time axis to find a stable region of the characteristic value. This stable region is called a window. An approach which fixes the outside of the stable region to zero to determine an elution profile for each component is particularly called WFA (Window Factor Analysis). For details of WFA, see E. R. Malinowski, J. Chemometrics, No. 6, pp. 29-40 (1992); and H. R. Keller et al., Anal. Chim, Acta., No. 246, pp. 379-390 (1991).
3-3. RAEFA (Rank Annihilation by Evolving Factor Analysis): A method which iterates the peak resolution performed by EFA (3-2) while subtracting components one by one from a data matrix.
3-4. RAFA Using Information Entropy as Index: A method which ranks down a matrix for each component, on the basis of minimum information entropy when obtaining an elution profile (see I. Sakura et al., J. Chromatogr, No. 506, pp. 223-243 (1990)).
However, although peak separation or resolution methods relying on the above-mentioned conventional leastsquares method are resistant to noise and suitable for practical use, a model function of any form must be introduced for the peak separation or resolution, so that it cannot be said that these methods always reproduce correctly actual peak shapes. Stated another way, while a model function is indispensable for the non-linear least-squares method, it is doubtful whether an employed model function is appropriate or not. Thus, the model function does not always give best-fit to actual peak shapes.
The problem of appropriateness also exists in the chromatographic quantitative analyzing method and apparatus described in JP-A-60-24447, and it cannot be said that the synthesized component function fs(l), employed therein, always correctly reproduces actual peak shapes, as is the case of the above-mentioned peak separation or resolution methods.
On the other hand, the deconvolution represented by Jansson""s""s method is not resistant to noise. Introduced noise would be largely amplified during the deconvolution to cause pseudo-peaks to appear in a resultant graph. Thus, even if the conventional deconvolution is applied to the analysis on a three-dimensional multichannel chromatogram, noise components cannot be removed, so that accurate peak separation or resolution is hindered.
The above-mentioned factor analysis based method for separating or resolving overlapping peaks on a multichannel chromatogram, in turn, has the following problems.
1. For employing the method with known spectrum, a standard spectrum need be known. This method cannot be applied if the spectrum is not known.
2. Since the modeling method with unknown spectrum requires a model function to be introduced for analysis, actual elution profiles are not always reproduced correctly.
3. The self modeling methods with unknown spectrum respectively imply various problems. Specifically, ITTFA in 3-1 and RAFA using information entropy as an index in 3-4 require a high resolution for overlapping peaks in a measured data matrix (see J. K. Strasters et al, Journal of Liquid Chromatography, No. 12 (1 and 2), pp. 3-22 (1989), I. Sakura et al). With a low resolution, these methods cannot ensure a meaningful solution because the calculation does not converge.
EFA in 3-2 and RAEFA in 3-3, on the other hand, are methods which assume that a region in which an elution profile presents zero is placed external to a window. Strictly speaking, however, the elution profiles do not include a zeroregion, although they may present values close to zero, so that the resolved result will include errors due to the assumption. In addition, it is difficult to clearly define a boundary between a zero-region and a non-zero region in practice.
It is therefore an object of the present invention to provide methods and apparatuses for analyzing a multichannel chromatogram for highly accurately resolving overlapping peaks on a multichannel chromatogram to provide information useful in analyzing the components of a sample.
To achieve the above object, according to a first aspect, the present invention provides a method for analyzing a multichannel chromatogram for analyzing chromatogram data having three-dimensional components composed of a characteristic component, a wavelength component, and a time component of a sample under measurement detected by a detector, which comprises the steps of compressing the chromatogram data having the three-dimensional components in terms of the wavelength component to transform to two-dimensional chromatogram data; performing deconvolution for removing a dispersion data portion due to an inherent device function representing the detection characteristic of the detector from the compressed two-dimensional chromatogram data; and identifying the characteristic component of the sample under measurement from the two-dimensional chromatogram data from which the dispersion data portion has been removed.
According to a second aspect, a method for analyzing a multichannel chromatogram of the present invention comprises the steps of compressing chromatogram data having three-dimensional components in terms of a wavelength component to transform the three-dimensional chromatogram data to two-dimensional chromatogram data; deconvoluting the compressed two-dimensional chromatogram to remove therefrom a dispersion data portion due to a normal distribution function which is an inherent device function of the detector representing its detection characteristic, and changing standard deviation of the normal distribution function to isolate overlapping peaks in the two-dimensional chromatogram data; performing convolution using the device function for adding dispersion data portions to respective components of the two-dimensional chromatogram data from which the dispersion data portion has been removed, to restore the data before the deconvolution; computing spectral information on the respective components based on the restored data; identifying the characteristic component of the sample under measurement from the computed spectral information and quantifying the characteristic component; and displaying at least the identified and quantified characteristic component and the restored data obtained by the convolution on display means.
According to a third aspect, a method for analyzing a multichannel chromatogram comprises the steps of performing deconvolution for removing a dispersion data portion due to an inherent device function representing the detection characteristic of a detector from three-dimensional chromatogram data; compressing the deconvoluted chromatogram data in terms of the time component to compute spectral information on the respective components; and identifying the characteristic component of the sample under measurement based on the computed spectral information.
According to a fourth aspect, an apparatus for analyzing a multichannel chromatogram for analyzing chromatogram data having three-dimensional components composed of a characteristic component, a wavelength component, and a time component of a sample under measurement detected by detector means, comprises a data compression unit for compressing the chromatogram data having the three-dimensional components in terms of the wavelength component to transform the three-dimensional chromatogram data to two-dimensional chromatogram data; a deconvolution unit for performing deconvolution for removing a dispersion data portion due to an inherent device function representing the detection characteristic of the detector means from the compressed two-dimensional chromatogram data; and a component identification unit for identifying the characteristic component of the sample under measurement from the two-dimensional chromatogram data from which the dispersion data portion has been removed.
Preferably, in the methods and apparatus for analyzing a multichannel chromatogram, the deconvolution unit or the deconvolution step changes standard deviation of a normal distribution function to isolate overlapping peaks in the two-dimensional chromatogram data.
Also, preferably, the methods and apparatus for analyzing a multichannel chromatogram further comprise a reconvolution unit and step for performing convolution using the device function for adding dispersion data portions to respective components of the two-dimensional chromatogram data from which the dispersion data portion has been removed, to restore the data before the deconvolution; and a display unit and step for displaying the identified characteristic component and the restored data obtained by the convolution, respectively.
According to another aspect, an apparatus for analyzing a multichannel chromatogram comprises a data setting unit for setting a start point and an end point for a time component of three-dimensional chromatogram data to be analyzed; a data compression unit for compressing chromatogram data having three-dimensional components set by the data setting unit in terms of the wavelength component to transform the three-dimensional chromatogram data to two-dimensional chromatogram data; a deconvolution unit for performing deconvolution for removing a dispersion data portion due to a normal distribution function which is an inherent device function representing the detection characteristic of a detector means, and changing standard deviation of the normal distribution function to isolate overlapping peaks in the two-dimensional chromatogram data; a separation/normalization unit for separating the deconvoluted chromatogram data into the respective components and normalizing the respective components; a reconvolution unit for executing convolution for adding a dispersion data portion to each of the components of the separated and normalized data to restore the data before the deconvolution; a spectrum computation unit for computing spectral information on the respective components based on the restored data; a component identification/quantification unit for identifying the characteristic component of the sample under measurement from the computed spectral information and quantifying the characteristic component; and display unit for displaying at least the identified and quantified characteristic component and the restored data obtained by the convolution.
Preferably, in the methods and apparatuses for analyzing a multichannel chromatogram, the display unit and step display the identified and quantified characteristic component, the compressed two-dimensional chromatogram data, the data on the respective components of the convoluted data, and the spectral information on the respective components.
Also preferably, in the methods and apparatuses for analyzing a multichannel chromatogram, the characteristic component of the sample under measurement is the absorbance.
According to a further aspect, an apparatus for analyzing a multichannel chromatogram comprises a data setting unit for setting a start point and an end point for each of a time component and a wavelength component of three-dimensional chromatogram data to be analyzed; a deconvolution unit for performing deconvolution for removing a dispersion data portion due to an inherent device function representing the detection characteristic of detector means from the three-dimensional chromatogram data to which the start points and the end points have been set; a normalized spectrum computation unit for normalizing the deconvoluted chromatogram data, compressing the normalized chromatogram data in terms of the time component, and computing spectral information on the respective components; and a component identification/quantification unit for identifying the characteristic component of the sample under measurement and quantifying the characteristic component.
Preferably, in the methods and apparatuses for analyzing a multichannel chromatogram, the device function is a normal distribution function.
Also preferably, the methods and apparatuses for analyzing a multichannel chromatogram further comprise display means and step for displaying the identified characteristic component and the spectral information on each component, respectively.
Also preferably, the methods and apparatuses for analyzing a multichannel chromatogram further comprise input means and step for inputting a region to be analyzed in the three-dimensional chromatogram data and a number of overlapping peaks to the data setting unit, respectively.
Also preferably, the methods and apparatuses for analyzing a multichannel chromatogram further comprise input means and step for inputting standard deviation of the device function, a plurality of retention times to be estimated, and an initial value of a fitting parameter for a time constant for specifying a tailing peak to the deconvolution unit.
According to a further aspect, a method for analyzing overlapping peaks on a multichannel chromatogram comprises the steps of deconvoluting the multichannel chromatogram using a predetermined dispersion function; performing multivariate analysis from the deconvoluted multichannel chromatogram; determining the width of the predetermined dispersion function and the deconvoluted multichannel chromatogram corresponding thereto in accordance with a criterion requiring that the peaks are isolated; and acquiring a spectral waveform from the determined multichannel chromatogram.
According to a further aspect, a method for analyzing overlapping peaks on a multichannel chromatogram comprises the steps of: deconvoluting the multichannel chromatogram using a predetermined dispersion function; factor-analyzing the deconvoluted multichannel chromatogram; determining whether a predetermined variate of the multichannel chromatogram is stable from the relationship between the width of the predetermined dispersion function and a resolution of the factor analysis corresponding thereto; determining the width of the dispersion function and the deconvoluted multichannel chromatogram in accordance with a criterion requiring that the peaks are isolated; and acquiring a spectral waveform from the determined multichannel chromatogram.
According to a further aspect, a method for analyzing overlapping peaks on a multichannel chromatogram comprises the steps of deconvoluting the multichannel chromatogram using a predetermined dispersion function; factor-analyzing the deconvoluted multichannel chromatogram; determining whether factor analysis corresponding to the width of the predetermined dispersion function uniquely has a solution; determining the width of the dispersion function and the deconvoluted multichannel chromatogram corresponding thereto in accordance with a criterion requiring that the factor analysis uniquely has a solution; and acquiring a spectral waveform from the determined multichannel chromatogram.
According to a further aspect, a method for analyzing overlapping peaks on a multichannel chromatogram comprises the steps of deconvoluting the multichannel chromatogram using a predetermined dispersion function; solving an eigenvalue problem from the deconvoluted multichannel chromatogram; determining whether the peaks are isolated from the relationship between the width of the dispersion function and a solution of the eigenvalue problem corresponding thereto; determining the width of the dispersion function and the deconvoluted multichannel chromatogram corresponding thereto in accordance with a criterion requiring that the peaks are isolated; and acquiring elution profiles from the determined multichannel chromatogram.
According to a further aspect, a data processing apparatus for analyzing overlapping peaks on a multichannel chromatogram comprises a first memory for storing the width of a dispersion function; a convolution unit for convoluting the dispersion function stored in the first memory; a factor analysis unit for factor-analyzing the deconvoluted multichannel chromatogram; a second memory for storing a solution of an eigenvalue problem of the factor-analyzed multichannel chromatogram; and an output unit for outputting a spectral waveform acquired from the deconvoluted multichannel chromatogram using the width of the dispersion function determined on the basis of the solution of the eigenvalue problem.
According to a further aspect, a data processing apparatus for analyzing overlapping peaks on a multichannel chromatogram comprises a first memory for storing the width of a dispersion function; a convolution unit for convoluting the dispersion function stored in the first memory; a factor analysis unit for factor-analyzing the deconvoluted multichannel chromatogram; a second memory for storing a solution of an eigenvalue problem of the factor-analyzed multichannel chromatogram; and an output unit for outputting a spectral waveform acquired from the deconvoluted multichannel chromatogram using the width of the dispersion function determined on the basis of the solution of the eigenvalue problem.
In a multichannel chromatogram having three-dimensional components composed of a characteristic component of a sample under measurement, a wavelength component, and a time component, noise components included in the chromatogram negatively and positively fluctuate, i.e., increase and decrease in the wavelength component direction. If the multichannel chromatogram is compressed in the wavelength component direction, the noise components will cancel with each other. For this purpose, the multichannel chromatogram is compressed in terms of the wavelength component to be transformed into two-dimensional chromatogram data. This transformation removes the noise components. The two-dimensional chromatogram data having the noise components removed therefrom is subjected to the deconvolution for removing data dispersion due to the device function, thus accurately resolving overlapping peaks. Then, the components of the sample under measurement are identified based on the accurately resolved data.
The noise components included in the multichannel chromatogram having the three-dimensional components also negatively and positively fluctuate, i.e., increase and decrease in the time component direction. Therefore, the multichannel chromatogram is first subjected to the deconvolution. Then, the deconvoluted data is compressed in terms of the time component for each of the components, and spectral information is calculated for each component. The noise components cancel with each other in this process and they are thus removed from the data. Finally, the components of the sample under measurement are identified based on the spectra having the noise components removed therefrom.
A basic procedure of DFA will be next described.
1. A data matrix D is deconvoluted. The deconvolution is intended to deconvolute a dispersion function h from the data matrix D to obtain a deconvoluted data matrix d, as represented by the following equation (16):
Deconvolution
Dij=hi("sgr")*dij("sgr")xe2x80x83xe2x80x83(16)
In the above equation, convolution of d with h results in the data matrix D. Replacing the discretely expressed time j by a continuous variable t and representing the deconvolution in an integration form, the following equation (17) is obtained. Note that convoluted matrices are represented in lower case letters:                                           D            i                    ⁡                      (            t            )                          =                              ∫                          -              ∞                        ∞                    ⁢                                    h              ⁡                              (                                                      t                    xe2x80x2                                    ;                  σ                                )                                      ⁢                          xe2x80x83                        ⁢                                          ⅆ                i                            ⁢                              (                                                      t                    -                                          t                      xe2x80x2                                                        ;                  σ                                )                                      ⁢                          ⅆ                              t                xe2x80x2                                                                        (        17        )            
Since the deconvolution is executed with the time t specified as a variable, the spectrum is processed for each channel with index i. Stated another way, each row vector of the matrix D is deconvoluted with the same dispersion function h.
It should be noted that in the case of chromatograms, Gaussian as expressed by the following equation (18) is normally used for the dispersion function h. Also, it is well known in the art to convolute a chromatogram using Gaussian as a dispersion function for resolving peaks.
h(t;"sgr")=(1/({square root over (2+L xcfx80)}"sgr"))exe2x88x92(t/2"sgr"2) xe2x80x83xe2x80x83(18)
The mechanism of the dispersion is described in detail in xe2x80x9cDYNAMICS OF CHROMATOGRAPHYxe2x80x9d by J. C. Giddings, published by Marcel Dekker, New York, 1965.
2. Prior to the deconvolution, standard deviation s serving as a criterion for the extension of Gaussian must be determined. Simply, selected for this case may be such standard deviation that permits all overlapping peaks to be separated and isolated. Actually, if all peaks are isolated in either of channels, the characteristic vector expressed by the equation (7) corresponds to each component, so that the elution profile matrix y can be determined.
The spectral matrix X can also be obtained in accordance with the following equations:
d=Xyxe2x80x83xe2x80x83(19)
x=dyxcfx84(yyxcfx84)xe2x88x921xe2x80x83xe2x80x83(20)
where y is an elution profile matrix isolated by the deconvolution. The elution profile matrix y is obtained by rotating and normalizing the characteristic vector when isolated.
Since the elution profile matrix Y can be determined in accordance with the following equation (21), i.e., since the matrix Y can be obtained from the matrices D and X, the resolution of overlapping peaks is completed:
Y=(XTX)xe2x88x921Xxcfx84Dxe2x80x83xe2x80x83(21)
As an alternative approach, at the time the matrix y is determined, the dispersion function h used in the convolution is used to reconvolute isolated peaks to restore the original elution profile matrix Y (this processing is hereinafter called xe2x80x9creconvolutionxe2x80x9d). Subsequently, the spectral matrix X may be computed in accordance with the equation (14).
3. Ideally, all peaks on a chromatogram are isolated by executing once the deconvolution. However, actually, the isolation of peaks implies the following problems: (1) the number of components included in overlapping peaks is unknown; and (2) Generally, when the deconvolution is performed with a dispersion function h having unnecessarily large standard deviation "sgr" , noise is extremely amplified.
RAFA can be applied for solving these problems.
Alternatively, the matrix D is deconvoluted as the standard deviation s is gradually extended to obtain matrices d("sgr"). Each of the matrices d("sgr") is subjected to factor analysis. As explained above, the transformation matrix T is generally indefinite. The transformation matrix has the nature of obtaining a physically meaningful elution profile matrix Y by orthogonal rotation transformation when peaks are isolated. Methods for finding out an appropriate orthogonally rotated transformation matrix T include xe2x80x9cvarimaxxe2x80x9d, xe2x80x9cquartimaxxe2x80x9d, and so on. For example, the matrices d("sgr") are sequentially factor-analyzed in the order of smaller "sgr". When a column vector in a obtained spectral matrix X presents a constant spectral waveform, determination can be made at this point that a peak is isolated. This constant column vector is determined as the spectrum of a first component, and the rank of the first component is degraded by one. Similar rank annihilation is repeated to complete the factor analysis for peak resolution.
Similarly, methods for determining oblique rotation (obliquemax, quartimin, biquartimin, covarimin, binormamin, maxplane, promax, and so on) may be used for factor analysis instead of orthogonal rotation. Alternatively, it is also possible to employ a criterion with which the characteristic vector becomes positive to the utmost, and the value of the following equation (22) becomes maximum:                               ∑          k                ⁢                              ∑            j                    ⁢                                                    "LeftBracketingBar"                                  ζ                  k                                "RightBracketingBar"                                      ⁢                          v              kj                                                          (        22        )            
Such a criterion has an advantage that the spectrum can be obtained even if peaks are not completely isolated, and small-scale deconvolution is sufficient to sequentially perform the rank annihilation.
In ordinary factor analysis, the spectral matrix X and the elution profile matrix Y cannot be uniquely obtained. The DFA employed by the present invention enables the matrices X and Y to be determined by performing deconvolution to isolate elution profiles corresponding to respective components. The essence underlying this approach is that the matrix D is the product of the matrices X and Y, and that when the matrix D is deconvoluted, the deconvolution actually acts only on the matrix Y. Since the matrix X is preserved when the elution profiles are isolated, a spectral waveform can be immediately generated. The matrix Y may be computed from the matrix D using the matrix X or may be obtained by reconvoluting isolated elution profiles. On the other hand, the deconvolution, when performed on a two-dimensional chromatogram, gives rise to a problem that pseudo-peaks are amplified. The DFA, however, extends its analysis to multichannel data so that noise components cancel with each other, thus eliminating the occurrence of pseudo-peaks. Further, the appropriateness of the processing itself can be evaluated by executing factor analysis simultaneously with the deconvolution. This evaluation is important since the present invention iteratively executes this processing. For example, when the matrix D is transformed to the matrix d, the matrix V is simultaneously transformed to the matrix v (abstract elution profiles after the processing). On the other hand, the matrix U remains invariant after the execution of this processing. As shown in the left half of FIG. 18, a matrix v obtained by solving the eigenvalue problem to obtain the matrix V from the data matrix D and deconvoluting the matrix V should be equal to a matrix v obtained by deconvoluting the data matrix D to obtain the matrix d and solving the eigenvalue problem for the matrix d. The appropriateness of the deconvolution can be evaluated by comparing these two matrices v. Alternatively, when a tolerable range is exceeded, the deconvolution may be corrected such that substantially tolerable results can be obtained from either of the paths, thus ensuring the matching of the two paths. The same can be applied to the right half of FIG. 18. By comparing two paths from the matrix V to the matrix y, the appropriateness of the deconvolution is evaluated and, if a tolerable range is exceeded, the deconvolution may be corrected to ensure the matching of the two paths. It should be noted however that since the transformation T is involved in this case, rotation over completely the same angle must be performed on the two paths. Generally, rotation R determined after the deconvolution is employed. By solving the eigenvalue problem with the approach described herein, the deconvolution can be corrected to be more appropriate such that pseudo peaks can be prevented to the utmost from occurring due to noise amplified by the deconvolution which has been a disadvantage of this processing.
Further, as shown in FIG. 19, the appropriateness of the entire DFA can be evaluated by examining whether a matrix y obtained by deconvoluting the matrix Y computed from the matrix D using the spectral matrix X (through the upper path) matches a matrix y obtained from the deconvoluted matrix d using the same matrix X (through the lower path).